The meaning of properties of XOR operation in $GF(2^n)$ In Cryptograph, I read some explanation about Exclusive-Or operation in Galois Field.

The five properties of the exclusive-or operation in the $GF(2^n)$
  field makes this operation a very interesting component for use in a
  block cipher: closure, associativity, commutativity, existence of
  identity, and existence of inverse.

I learned what $GF(2^n)$ is, but I cannot understand the meaning of these properties.
What these properties mean?
 A: I. Block cipher (slide source):



II. Exclusive or:


$A\neq B\implies1$



*

*In special when: $A=X$ $\land$ $B=\widetilde{X}\implies A\oplus B\Leftrightarrow X\oplus\widetilde{X}$ is always $1$.


The other properties for one (unknow) element $X$:


*

*$(X\oplus0)=X$

*$(X\oplus1)=\widetilde{X}$

For $n$ inputs we have a Galois field of $2^n$ elements $\iff \forall a_i=0 $ $\vee$ $a_i=1 $


III. XOR block cipher:


*

*Closure:
The field's addition operation of $GF(2)$ is corresponds to the logical XOR operation of 2 inputs

*Associativity: Here for $GF(8)$:
$$(A\oplus B)\oplus C= A\oplus (B \oplus C)=(A\oplus C )\oplus B
$$


*Communatativity: Short: it's true (table in point II. Also: $GF(4)$)

*Existance of ideantity:


Question: Does exist an element $E_1$ where: $E_1\oplus X\iff X\oplus E_1 = X$ ? Answer: Yes, for $E_1=0$


*

*Existance of inverse: $X\oplus\widetilde{X}=\widetilde{X}\oplus X=E_2=1$

IV. Summary:
XOR- This is sometimes thought of as "one or the other but not both". This could be written as "A or B, but not, A and B".
The field's addition operation for $GF(2^n)$ till each element equal $0$ or $1$  is corresponds to the logical XOR operation.
